The present disclosure relates to imaging of geophysical subsurface features such as sedimentary layers, salt anomalies, and barriers. Specifically, the present disclosure relates to an inversion technique that provides high-fidelity images of subsurface regions from geophysical data sets.
Geophysical inversion generally involves determining values of at least one geophysical parameter of a subsurface region (e.g., wave speed/velocity, permeability, porosity, electrical conductivity, and density) in a manner which is consistent with at least one geophysical data set acquired at locations remote from the region (e.g., seismic, time-lapse seismic, electromagnetic, gravity, gradiometry, well log, and well pressure measurements).
Geophysical inversion is often carried out through the minimization of an objective function which includes a least-squares data misfit and a regularization term (Parker, 1994). Such an objective function typically takes the formJ(κ)=∫M∥u(κ)−ū∥2dx+R(κ),  (1)where κ represents the unknowns to be determined through optimization, ū is the measured geophysical data, u(κ) is data computed during the optimization according to a mathematical model, M denotes the locations at which the data is measured, ∥·∥ is an arbitrary norm used to quantify the error or misfit between the computed data u(κ) and the measured data ū, and R(κ) is a regularization term (e.g. Tikhonov regularization) used to stabilize the inversion.
Although the true physical parameter κ is a spatially varying function defined at every point of a subsurface domain Ω, κ is typically discretized, for computational purposes, when optimizing the objective function J(κ). Such a discretization typically amounts to first partitioning the subsurface domain Ω into disjoint cells Ωi, such that Ω=υΩi, and then setting κ(x)=κi for x∈Ωi. One example of such a partitioning into cells is a regular Cartesian grid. The parameter values {κi} are determined by optimizing equation (1) using a gradient-based optimization technique such as Gauss-Newton, quasi-Newton, or steepest descent. Henceforth, this approach will be referred to as the cell-based (or cell-by-cell) inversion technique (Aster, 2013).
For certain problems, such cell-based inversion techniques fail to demonstrate robust, high-fidelity performance when imaging subsurface regions. This is often because the objective function is ill-posed or under-constrained, in which case there may not be a unique solution to the objective function or the solution may be very sensitive to noise in the acquired geophysical data set. Regularization is often included in equation (1) in order to address these issues, but state-of-the-art regularization methods may introduce into the inversion, artifacts such as unnecessary smoothing. Moreover, this non-uniqueness becomes more severe when inverting for multiple distinct physical parameters (e.g., wave speed and density), making inversion particularly problematic in such cases.
One possibility for reducing the non-uniqueness present in multi-parameter inversion is to require that different physical parameters be structurally consistent. Structural consistency of two parameters means that the parameters share a common set of interfaces across which they change abruptly. Many geologic parameters, such as wave speed and density, exhibit such a relationship. However, despite the prevalence of such examples, conventional inversion techniques, as described above, cannot ensure that such a criterion is satisfied.
Therefore, for geophysical inverse problems requiring the determination of values of multiple parameters, additional steps must be taken in order to ensure that inverted values for the parameters are structurally consistent. One approach for doing this is to link the different parameters through geophysical or empirical relationships (e.g., Archie's law) and invert for a minimal subset of the parameters. When well-known relationships between parameters are available, this approach has great appeal. However, such relationships are, in many cases, either poorly understood or unknown completely, rendering this approach of limited value.
Another approach, known as the cross-gradient approach, inverts for two distinct physical parameters by introducing an additional term into the objective function which penalizes gradients that are not collinear. As a result, the cross-gradient approach encourages structural consistency, but it does not strictly enforce it. In addition, it is unclear how this approach generalizes to an arbitrary number of different physical parameters.
A third inversion methodology, which does ensure structural consistency, uses level-set functions to represent interfaces between subsurface regions. This approach has been applied to inverse problems such as full-wavefield inversion, gravity inversion, magnetic inversion, and reservoir history matching. However, due to the nature of the level-set representation, it is unclear how effective this approach will be for problems with a large number of distinct subsurface regions that contain possibly heterogeneous parameter values.
In light of the above described drawbacks of existing geophysical inversion techniques, the inventors recognized a need in the art to develop alternative geophysical inversion techniques that overcome or improve upon such limitations.